Let $K$ be a knot in the boundary of a compact smooth 4-manifold $X$, and suppose that $K$ is the the kernel of $\pi_1(\partial X) \to \pi_1(X)$. Then $K$ is the boundary of some immersed disk $D \to X$ that has some fine number of double points. I am interested in knowing how to compute minimum number of such double points, minimized over all such immersed disks. For now, I will call it the immersion number of $K$ - denoted $I(K)$, for fun. Has this quantity been studied in the literature?

One thing to note about $I(K)$ is that it provides a lower bound on the minimum genus of an orientable surface bounding $K$ (the "4-ball genus" if $X = B^4$). I am interested in knowing examples of knots (even/especially with $X = B^4$) where $I(K)$ and the minimum genus of an orientable surface properly embedded in $X$ bounding $K$ are not equal.

  • $\begingroup$ It is maybe not of your interest. In case when $K$ is not a knot but essential circle in $\partial X$ which is trivial in $X$ then we can find embedding of disk $D\to X$ which bounds our circle, right ? This is famous Dehn lemma in 3-manifolds but in 4-manifolds I don't know whether it holds. My interest is following: for given $M^3$ find minimal $N^4 \supset M$ with trivial first homology. Intuition is that $M$ should cut $N$ onto two pieces. I wonder when we can find $N$ also with trivial second homology which makes it homology sphere. $\endgroup$ – user21230 Oct 11 '18 at 14:17

For knots in $S^3=\partial B^4$ this is called the four-ball crossing number or clasp number.

For knots in $S^3$ you have the following inequalities relating $I$ to the $4$-ball genus $g_4$ and the unknotting number $u$

$$g_4\leq I \leq u.$$

In general, these are no equalities. For counterexamples, you can have a look at the Table of Knot Invariants.

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  • $\begingroup$ Thank you this is exactly what I was looking for! $\endgroup$ – user101010 Oct 9 '18 at 19:09
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    $\begingroup$ For reference, the knot $9_35$ has $u=3, I=2, g_4=1$ - it is the lowest crossing number knot with all of Marc's inequalities strict. $\endgroup$ – user101010 Oct 9 '18 at 19:30

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